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Chapter 3: Q4E (page 129)

Suppose thatXandYhave a continuous joint distribution for which the joint p.d.f. is defined as follows:

Determine (a) the value of the constant c;


Short Answer

Expert verified

a. The value of the constant is 1.5.

b. The probability is 0.375.

c. The probability is 0.125.

d. The probability is 0.50.

e. The probability is 0.00.

Step by step solution

01

Given information

The pdf of X and Y is given by,


02

Finding the value of the constant

03

Calculating the probability for part (b)

04

Calculating probability for part (c)

05

Calculating the probability for part (d)

06

Calculating the probability for part (e)

e.

Here X and Y are continuous random variables.

So the probability of continuous random variable take specific value is 0.

Pr (X = 3Y)

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Most popular questions from this chapter

Question:LetYbe the rate (calls per hour) at which calls arrive at a switchboard. LetXbe the number of calls during at wo-hour period. A popular choice of joint p.f./p.d.f. for(X, Y )in this example would be one like

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{{{\left( {{\bf{2y}}} \right)}^{\bf{x}}}}}{{{\bf{x!}}}}{{\bf{e}}^{{\bf{ - 3y}}}}\;{\bf{if}}\;{\bf{y > 0}}\;{\bf{and}}\;{\bf{x = 0,1, \ldots }}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

a. Verify thatfis a joint p.f./p.d.f. Hint:First, sum overthexvalues using the well-known formula for thepower series expansion of\({{\bf{e}}^{{\bf{2y}}}}\).

b. Find Pr(X=0).

Suppose that the p.d.f. of X is as follows:

\(\begin{aligned}f\left( x \right) &= e{}^{ - x},x > 0\\ &= 0,x \le 0\end{aligned}\)

Determine the p.d.f. of \({\bf{Y = }}{{\bf{X}}^{\frac{{\bf{1}}}{{\bf{2}}}}}\)

Suppose that \({{\bf{X}}_{\bf{1}}}\;{\bf{and}}\;{{\bf{X}}_{\bf{2}}}\) are i.i.d. random variables andthat each of them has a uniform distribution on theinterval [0, 1]. Find the p.d.f. of\({\bf{Y = }}{{\bf{X}}_{\bf{1}}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}\).

Suppose that the n variables\({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}\) form a random sample from the uniform distribution on the interval [0, 1]and that the random variables \({{\bf{Y}}_{\bf{1}}}\;{\bf{and}}\;{{\bf{Y}}_{\bf{n}}}\) are defined as in Eq. (3.9.8). Determine the value of \({\bf{Pr}}\left( {{{\bf{Y}}_{\bf{1}}} \le {\bf{0}}{\bf{.1}}\;{\bf{and}}\;{\bf{Y}}_{\bf{n}}^{} \le {\bf{0}}{\bf{.8}}} \right)\)

Let Xbe a random variable with the p.d.f. specified in Example 3.2.6. Compute Pr(X≤8/27).
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